Multiquadric radial basis function approximation method and asymptotic numerical method for nonlinear and linear static analysis of single-walled carbon nanotubes

Abstract

In this paper, a numerical algorithm combining the meshless collocation technique based on the globally supported Multiquadric Radial Basis Function (MQRBF) approximation method and the Asymptotic Numerical Method (ANM) is developed to investigate the nonlinear and linear static behaviors of Single-Walled Carbone Nanotubes (SWCNTs) based on a Nonlocal Continuum Beam Model (NCBM). The NCBM which accounts for the effects of small scale, shear deformation and geometric nonlinearity of von-Kármán type is constructed by implementing the Nonlocal Elasticity Theory of Eringen (NET) into the First-Order Shear Deformation Elastic Beam Theory (FOSDEBT) and used to model the SWCNTs as a Continuous Elastic Nanobeam (CEN). Based on the Total Lagrangian Formulation (TLF) the nonlocal nonlinear governing equations of SWCNT are elaborated in a quadratic matrix strong form. These equations are then solved numerically by applying the proposed algorithm. This algorithm permits to transform the quadratic nonlocal nonlinear governing equations into a sequence of linear ones shared the same tangent stiffness matrix which are then solved numerically by the MQRBF approximation method, to get the whole branch of solution a continuation procedure is required. In order to highlight the reliability and efficiency of the present numerical algorithm, comparative studies are conducted.